A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Saturday, June 26, 2010

I haven't had much time to write, but I thought I'd send a postcard from this edge.

We've just finished three weeks of the 2010 REU, and things are chugging along smoothly.

Incredibly smoothly.

Wow.

The students' first presentations (yesterday) were masterful, already conference-worthy: their talk mechanics were spot-on, they displayed superb mastery of the concepts and computations they presented, and their slide designs were eminently professional. In every way their presentations were superior than many of those I've seen given at major conferences by seasoned mathematicians.

They're ready for the "minisymposium" we're playing host to next week, to which we've invited the faculty and students from two other area REUs.

I hope that I can take some of the credit for the students' success to this point: I've been yet more intentional in my design of the program's activities than I've ever been in the past, providing even more and more timely instruction on writing, presenting, use of scholarly sources, reading of math papers, etc. than in past iterations of the program.

And I've had help in this, too, that I've not had before: Bella and Damian, two of my Charleston-based partners in composition-theoretic crime, came up the mountain a little over a week ago to talk to the kids about writing and math. The visit was twofold...perhaps even threefold: (1) to give the students the rhetorician's point of view on mathematical writing, (2) to interview (though this term was never used) about their past experiences with academic writing in general and mathematical academic writing in particular, and (3) to get our shit together and make some progress on the paper we began back in February when I joined them for a working weekend down in South Carolina. (To this last point, briefly: I was tasked with coauthoring sections on the students' mastery of visual rhetoric, specifically with regard to the visual effects on the reader of white space and mathematical formulas; and on the students' use of sources. My analysis of the latter was very enlightening to me, and helped me quantify the fact that this year's students are well ahead of their predecessors when it comes to making effective use of the literature they find in the course of their investigations.)

The two-hour-long "interview" with the students was wonderful. In the discussions that arose the students revealed rich and robust histories with writing, including a good deal of relevant and highly intentional instruction I might not have expected. These students are very well-prepared (which helps to explain the success they've had so far in this summer's program!). The interview also gave us a great deal of data we can use as a foundation for our next article, in which we hope (already!) to trace the development of students as communicators of mathematics through the course of a summer research program one of whose foci is on writing in the discipline. Bella's already kicking around the idea of coming back toward the end of the program to interview the students again and see if anything's moved.

Mathematically, the students are fantastic. They're very self-directed and intrinsically motivated. They're displaying superlative creativity and originality of thought, to a greater extent than did many students in previous years of the program. They're making terrific use of the sources they find, plying them effectively to serve multiple purposes: to find other sources, to support their own propositions and theorems, and to situate their work in the context of what's come before. And every one of them has a clear sense of purpose in his or her project: no one of them is "stuck," as I've seen happen in the past.

More important, perhaps: they're getting along exceedingly well with each other. From Day One they've bonded remarkably well. They do almost everything together, not because they feel like they have to or because they're afraid to do thing alone (both of these have been moving forces in previous iterations of the program), but because they want to. They clearly like each other. A lot. It's good to see.

It comes across in simple things, too: yesterday, during the students' first round of talks, this year's group proved themselves to be the most attentive and responsive audience members we've yet had in the program. They listened with clear and unfeigned interest, they gave encouraging nonverbal feedback (nods and smiles) at appropriate moments, and they asked a number of good questions of their peers at the end of every talk. Midway through the talks Tip (my sole faculty partner in running the program this year) and I marveled at the fluency of the students' presentation skills and remarked at how though the students had all chosen to work on individual projects (something we initially bemoaned) it was evident that to a greater extent than in previous years the students are collaborating with one another behind the scenes, sharing ideas, talking about LaTeX and Beamer, and helping each other through rough spots.

Ladies and gentlemen, we have established an effective learning community.

What else is up?

I've only been to one meeting of the Learning Circle (having missed last week's to work the run-off election) I chose to take part in this summer, but I have a good feeling about the direction it seems to be taking.

I've received good feedback on the proposal I sent to Jossey-Bass for a text on writing in mathematics.

I've finally submitted a reasonable draft of an assessment plan for the university's Writing Intensive student learning outcomes to the powers that be. I doubt that's the last I'll hear of that, though.

It's been a busy summer, but a good one. I'm excited to see what the coming week will bring.

For now, I'm going to kick back and watch Ghana get the better of the U.S. soccer team.

Tuesday, June 08, 2010

We've wrapped up two days of the 2010 REU, and things are running smoothly so far.

The students are gelling tremendously quickly into a friendly and supportive learning community. As early as the welcome barbecue a few nights back it was obvious these folks were going to get along really well, and the last couple of days have borne that observation out. "I was a little worried before coming here," one of the students told me at the barbecue, "but everyone's so nice that I know I'm not going to have trouble working with them. It's going to be easy to ask them for help if I need it."

Although I can't yet claim to know them all incredibly well as individuals, each is beginning to make known her or his particular strengths and weaknesses. Three of them are particularly eager to demonstrate their skills and hesitate not at all in traipsing to the board to show off their solutions. A couple of the others are more retiring, willing to whisper the answers (almost always correct) from where they sit but less enthusiastic about taking to the board. The remaining three are quieter still and it often takes a little prodding to get them to share their ideas. Notably, to an extent not seen since the first year of the program, one of the students is clearly emerging as the "social director." She's the most outgoing of the group, though a couple of her peers aren't far behind when it comes to eagerness and enthusiasm.

I should note that the students needed little convincing to complete the first day's freewriting exercise, which was designed to tease out of the students their personal goals for the overall program. I've posted the resulting list of student goals here; happily, most of the learning goals coincide very well with my own.

The students are making quick progress through the long list of graph theoretical definitions I bludgeoned them with yesterday afternoon, and we're already over halfway through the graph theory notes I hope to complete before calling it quits. We've also hit on several topics from dynamical systems and geometry, and tomorrow brings more graph theory, dynamics, LaTeX, and Mathematica. It's going to be a full day, but I think we'll get through it.

Further bulletins as events warrant. For now, bed calls. My bedtime reading? The basis for the Learning Circle in which I'm taking part this summer, Helping college students find purpose: the campus guide to meaning-making, by Robert J. Nash and Michele C. Murray (2010, San Francisco: Jossey-Bass). We'll see what comes of it.

Sunday, June 06, 2010

As of 11:00 or so last night, all of the 2010 REU students are in town.

They're gelling as a group really well this year, much as they did two years ago. Last year's folks took a little while to come out of their shells, although once they did so, they got along famously and I joy in seeing their friendships continue via Facebook.

This year's crew is ready to get underway, I think. They're as excited as I am to get started tomorrow morning.

As I've mentioned before, I've made some changes, mostly minor, and all for the best, I hope, in the program this year.

What's new?

1. New focus on specific aspects of communication. As I indicated in my last post, I'll be requiring students to practice "elevator talks" and provide at least one visual in their every-other-weekly presentations. My hope is that these new requirements will lead to greater intentionality on the part of the students as they craft their talks and papers.

2. Deepening of engagement with source material. Students will now be required to perform a thorough literature search as a natural adjunct of their respective research projects, and this literature search will be accompanied by the creation of a simple annotated bibliography. This exercise should help them in developing meaningful "history" sections in their papers.

3. Alternation of presentations and papers. The students will be completing only four (not six) drafts of their papers throughout the summer, at the end of the second, fourth, sixth, and eighth weeks. They will be presenting only every other week, rather than every week, at the end of the third, fifth, seventh, and eighth weeks. We will still have a mandatory meeting on every Friday, but on alternating Fridays this time will be used for conferencing on papers, peer review, practice of elevator talks, and so forth.

4. Emphasis on collaboration. In addition to more frequent peer review (I plan to have them perform three peer reviews, as opposed to the one done in the past years), I'll be emphasizing the collaborative and communal nature of mathematical research by encouraging them to work together on their projects. As exciting as it is to have eight students working on eight different projects, the students stand to gain more by working with one another, and my attention can be much better focused if I only need to stay on top of three or four projects rather than eight.

I've got a good feeling about it. They're a bright bunch, they're getting along well already, and I think it's going to be a particularly good summer.